Abstract In this work we give a proof of the mean-field limit for λ-convex potentials using a purely variational viewpoint. Our approach is based on the observation that all evolution equations that we study can be written as gradient flows of functionals at different levels: in the set of probability measures, in the set of symmetric probability measures on N variables, and in the set of probability measures on probability measures. This basic fact allows us to rely on Γ-convergence tools for gradient flows to complete the proof by identifying the limits of the different terms in the Evolutionary Variational Inequalities (EVIs) associated to each gradient flow. The λ-convexity of the confining and interaction potentials is crucial for the unique identification of the limits and for deriving the EVIs at each description level of the interacting particle system.

中文翻译：

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弱相互作用随机粒子混沌传播的基于λ凸性的证明

摘要 在这项工作中，我们使用纯变分观点给出了 λ 凸势的平均场极限的证明。我们的方法基于以下观察：我们研究的所有进化方程都可以写成不同级别的函数梯度流：在概率测度集合中，在 N 个变量的对称概率测度集合中，以及在概率集合中概率测度的测度。这一基本事实使我们能够依靠梯度流的 Γ-收敛工具通过识别与每个梯度流相关的进化变分不等式 (EVI) 中不同项的限制来完成证明。